Well, augmenting reality with an extra dimension containing the thing that previously didn’t exist is the same as “trying and seeing what would happen.” It worked swimmingly for the complex numbers.
No it isn’t. The difference between i and the values returned by Maverick() is that i can be used to prove further theorems and model phenomena, such as alternating current, that would be difficult, if not impossible to model with just the real numbers. Whereas positing the existence of Maverick() is just like positing the existence of a finite value h that satisfies h=10. We can posit values that satisfy all kinds of impossibilities, but if we cannot prove additional facts about the world with those values, they’re useless.
You can prove additional facts about the world with those values, that was the point of my usage of ‘i’ as one of the examples.
For h = 1⁄0 you can upgrade R to the projectively extended real line. If I’m not mistaken, one needs to do this in order to satisfy additional proofs in real analysis (or upgrade to this one instead).
You seem to be asking whether or not doing so in every conceivable case would prove to be useful. I’m saying that we’d be likely to know beforehand about whether or not it would be. Like finding polynomial roots, one might be inclined to wish that all polynomials with coefficients in the reals had roots, therefore, upgrading the space to the complex numbers allows one to get their wish.
No it isn’t. The difference between i and the values returned by Maverick() is that i can be used to prove further theorems and model phenomena, such as alternating current, that would be difficult, if not impossible to model with just the real numbers. Whereas positing the existence of Maverick() is just like positing the existence of a finite value h that satisfies h=10. We can posit values that satisfy all kinds of impossibilities, but if we cannot prove additional facts about the world with those values, they’re useless.
You can prove additional facts about the world with those values, that was the point of my usage of ‘i’ as one of the examples.
For h = 1⁄0 you can upgrade R to the projectively extended real line. If I’m not mistaken, one needs to do this in order to satisfy additional proofs in real analysis (or upgrade to this one instead).
You seem to be asking whether or not doing so in every conceivable case would prove to be useful. I’m saying that we’d be likely to know beforehand about whether or not it would be. Like finding polynomial roots, one might be inclined to wish that all polynomials with coefficients in the reals had roots, therefore, upgrading the space to the complex numbers allows one to get their wish.